Lecture 2 Consumer theory (continued)

Transcription

1 Lecture 2 Consumer theory (continued) Topics 1.4 : Indirect Utility function and Expenditure function. Relation between these two functions. mf620 1/2007 1

2 1.4.1 Indirect Utility Function The level of utility when x(p,y) is chosen must be the highest one for given prices and income. Different Prices and income will give us the different level of utility. Let call the function that relate the maximized utility to prices and income as the indirect utility function mf620 1/2007 2

3 1.4.1 Indirect Utility Function v(p,, y)= u(x(p x(p,y)) When u is continuous, x exist, then v is well-defined. From graph, v(p,y) gives the value of utility of the highest indifference curve that is tangent to the budget line. mf620 1/2007 3

4 x2 u=v(p1,p2,y) x1

5 1.4.1 Indirect Utility Function: Properties 1. Continuous at all p, y >0 2. Homogenous of degree zero in (p, y) 3. Strictly increasing in y 4. Decreasing in prices 5. Quasiconvex in (p, y) mf620 1/2007 5

6 1.4.1 Indirect Utility Function: Properties 6. Roy s s Identity: x 1 (p, y) = - v/ v/ p 1 / v/ v/ y Proof 1. By the theorem of the Maximum 2. To show that v(tp tp,ty)=t 0 v(p,y). Intuitively, the budget line is not shifted. 3. v/ v/ y y > 0. Using the envelop theorem, we know this is equal to λ * and > 0. Intuitively, more income leads to higher utility. mf620 1/2007 6

7 1.4.1 Indirect Utility Function: Properties 4. v/ v/ p p < 0. Use envelope theorem. 5. By definition v(p t,y t ) max[v(p 1,y 1 ), v(p 2,y 2 )] Consumers prefers one of any two extreme budget sets to any average of the two. mf620 1/2007 7

8 1.4.1 Indirect Utility Function: Properties Or set {p: v(p,y) k} or lower contour set is a convex set. p2 p1 mf620 1/2007 8

9 1.4.1 Indirect Utility Function: Properties v v(p,y) y mf620 1/2007 9

10 1.4.1 Indirect Utility Function: Properties 6. Roy s s identity, from v/ v/ p 1 = - λ * x * 1 And v/ v/ y y = λ * mf620 1/

11 1.4.1 Indirect Utility Function: Properties Let check all properties from familiar utility. Using CD utility functions v(p1,p2,y) = y / p a 1 p 1-a 2 Try CES function as shown in p v(p1,p2,y) = y / (p r ( 1 + p r 2 ) -1/r mf620 1/

12 1.4.2 Expenditure Function What is the minimum level of money or expenditure needed for given prices to get a given level of utility? min x px st.. u(x) u. We specify the attainable level of u, then ask for the minimum money of achieving it. mf620 1/

13 x2 Isoexpenditure: e=p1x1+p2x2 Slope=-p1/p2 u e2/p1 e3/p1 x1

14 1.4.2 Expenditure Function min x px st.. u(x) u. Let x h (p,u) solves the problem The lowest expenditure to get u is equal to px h (p,u) Let e(p,u) px h (p,u) Called the Expenditure function. mf620 1/

15 1.4.2 Expenditure Function x h (p,u) is called Hicksian Demand or Compensated demand function. We can draw the Hicksian demand for good1 by letting the price of x1 to vary, as we did for the Marshallian demand. mf620 1/

16 x2 Let p 1 changes from p 10 to p 1 1 Before: slope= - p 10 / p 2 0 a b u After: slope= - p 11 / p 2 0 x1

17 p1 p 1 0 We assume p2 and u constant: Hicksian demand for good 1 p 1 1 x1(p1, p 20,u) X 1 at a X 1 at b x1

18 1.4.2 Expenditure Function Hicksian demand tells us what consumption bundles achieves a targeted utility and minimizes total expenditure. Compensated term means as price of good 1 varies (goes up), you lose some utility, so to keep u constant, you needed to be compensated to be as happy as before. Hicksian demand is not directly observable since it depends on utility, mf620 1/ which is not observable.

19 1.4.2 Expenditure Function: properties Theorem Continuous in p 2. Strictly increasing in u 3. Increasing in p 4. Homogenous of degree 1 in p 5. Concave in p mf620 1/

20 1.4.2 Expenditure Function: properties 6. Shepard s s Lemma x h 1 (p,u) = e(p,u)/ e(p,u)/ p 1 Proof. (try it by yourself, same ideas as in the indirect utility) mf620 1/

21 1.4.2 Expenditure Function: properties Example Min p1x1 + p2x2 St. u(x1, x2) =u L(x1,x2,λ) ) = p1x1 + p2x2 [u(x1, x2) u] Suppose u(x1, x2) =x 1a x 1-a 2 answer x h 1 (p,u) = a(p 2 / p 1 ) 1-a u x h 2 (p,u) = (1-a)(p 1 / p 2 ) a u And e(p,u) = p a 1 p 1-a 2 u Check Shepard s s lemma Do CES in example1.3 p. 38 mf620 1/

22 1.4.3 Relations between Expenditure and Indirect utility If you did my assignment, you know that v(p1,p2,y) = y / p a 1 p 1-a 2 If we compare this with the expenditure function e(p,u) = p a 1 p 1-a 2 u We see that the expenditure function is the inverse of indirect utility function. We can inverse v function to see y that relates to u and p. mf620 1/

23 Theorem 1.8 Duality between indirect utility and expenditure If we replace y with e, and v with u in the indirect utility function we will get the expenditure function, and vice versa. 1. e(p, v(p,y) ) = y The minimum expenditure necessary to get utility v(p,y) is y. 2. v(p, e(p,u) ) = u The maximum utility from income e(p,u) is u. mf620 1/

24 Theorem 1.8 Duality between indirect utility and expenditure This duality lets us work with one problem and invert them to get solution of another problem. mf620 1/

25 y/p2 x2 Maximized utility and minimized expenditure (x 1*,x 2* ) u(x * )=u x1 y/p1 = e/p1

26 Theorem 1.9 Duality between Marshallian and Hicksian Demand If x* solves UPM, let u=u(x*), then x* solves EM. Where UPM: max x u(x) st px y And EM: min x px st u(x) u. Similarly, If x* solves EM, let y =px*, then x* solves UPM. mf620 1/

27 Theorem 1.9 Duality between Marshallian and Hicksian Demand 1. x (p, y) = x h (p, v(p,y) ) The Marshallian demand at income y is the same as Hicksian demand at utility v(p,y) 2. x h (p,u) = x (p, e(p,u) ) Hicksian demand at utility u is the same as the Marshallian demand at income e(p,u). mf620 1/

28 p1 x 1 x h 1 x 1* =x 1h (p,u)= x 1h (p,v(p,y))=x 1 (p,y) x1

29 UMP EM x(p,y) x h (p,u) v(p,y) e(p,u) x 1 (p, y) = - v/ v/ p 1 / v/ v/ y x h 1 (p,u) = e(p,u)/ e(p,u)/ p 1